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Daker Regiment
Notation used here: \(\hat{\text{R}}\) is defined in the John Daker article. \(A(\alpha)=\hat{\text{R}}_{\alpha}^{10\uparrow 10\uparrow 100}(10\uparrow 10\uparrow 100)\) \(\hat{\phi}\) is the Hyperordinal Veblen Phi Function. \(\{\alpha,\beta\mapstoFunction\}\{Array\}\) is a transfinite extension of Customizable BEAF. \(g^{(a_1,a_2,\cdots)}(b_1,b_2,\cdots)\) is Multiargument Iteration Notation. \(\dot\vartheta(\{a\mapsto Function\})\) denotes the growth rate of the function on the FGH. \(\dot\vartheta(b)=\min\{\alpha:\forall x:f(\alpha,x)\ge b(x)\}\) \(\hat\vartheta(\{a\mapsto Function\})\) denotes the set-theoretical order of a function (defined below). Orders of Functions Define the order of a function as the least order of set theory needed to define/approximate the function. For example, the Graham Function has order 1 because it can be defined using only FOST, but the FOST Function has order 2 because a full definition would require 2nd-order set theory. Multiargument functions can also have orders as well. Order Proof First, assume that the function has an order. \(\hat{\vartheta(\hat{\text{R}}})\) would be the order required to define the \(\hat{\text{R}}\) function mentioned above. If the \(\hat{\text{R}}\) function has an order, then how would \(\hat{\text{R}}(\hat{\vartheta(\hat{\text{R}}})+2,n)\) wouldn't be definable because it requires the R function to generalize over an order greater than the function's own order. But if the ordinal \(\hat{\vartheta(\hat{\text{R}}})\) exists, then \(\hat{\vartheta(\hat{\text{R}}})\)-order set theory must as well, so the \(\hat{\text{R}}\) function must not be definable in set theory of any order. Notations/Applications The \(\hat{\hat{\hat{\text{R}}}}^{\vdots}_\alpha(n)\) function with \(m\) hats can be denoted as \(\hat{\text{R}}_{\alpha,m}(n)\). Further Extensions/Higher Oracles \(\hat{\varepsilon}\) Oracles Define a second order \(\hat{\hat{\text{R}}}\) function that has an oracle referred to as \(\hat{\varepsilon}_2\)The reason that this is \(\hat{\varepsilon}_2\) instead of \(\hat{\varepsilon}_1\) or \(\hat{\varepsilon}\) is so that the \(c\)th function uses the \(\hat{\varepsilon}_c\) oracle., which is a truth predicate of the language of any \(\hat{\text{R}}_\alpha\) function for any ordinal \(\alpha\) of any sizeThere is no limit on ordinal size in the definition. When the function is called, the length of the expression becomes a limit for the size.. The above proof can be repeated here to show that the \(\hat{\hat{\text{R}}}\) function is not definable in set theory even using the \(\hat{\varepsilon}_2\) oracle. This allows for the definition of the \(\hat{\hat{\hat{\text{R}}}}\) function with an \(\hat{\varepsilon}_3\) oracle used as a truth predicate for the \(\hat{\hat{\text{R}}}\) function's language. The oracle would work like this (except in Polish Notation and with Godel-encoding): \(\hat{\varepsilon}(1,c,\phi,a)\) would act as a truth predicate for an expression \(\phi\) and a variable assignment \(a\) in the language of \(\hat{\text{R}}_{c}(n)\) for \(c<\omega\). This can be extended to the \(\hat{\text{R}}_{\alpha,\omega}(n)\) function, which has access to an \(\hat{\varepsilon}_\omega\) oracle which in turn has access to any \(\hat{\varepsilon}_b\) oracle where the \(b\) is a FOST-encoded Von Neumann Ordinal. \(\hat{\varepsilon}_{\omega+1}\), \(\hat{\varepsilon}_{\omega+2}\), etc. can be defined similarly to the previous finite oracles by defining each as a truth predicate for the previous function's language. Similar to how \(\alpha\) order set theory has a \(\psi(\alpha,\phi)\) predicate able to access \(\psi(\beta,\phi)\) for \(\beta<\alpha\), \(\hat{\varepsilon}_\alpha\) is able to access the language used in \(\hat{\text{R}}_\beta\) for \(\beta<\alpha\). \(\hat{\zeta}\) Oracles \(\hat{\zeta}_2\) is a truth predicate for language including \(\hat{\varepsilon}_\alpha\) for any ordinal \(\alpha\). This is defined similarly to the way that \(\hat{\varepsilon}_2\) is a truth predicate for \(\hat{\text{R}}_\alpha\) with any \(\alpha\). \(\hat{\eta}\) Oracles This is a truth predicate for language including \(\hat{\zeta}_\alpha\) for any ordinal \(\alpha\). These start at \(\hat{\eta}_2\). \(\hat{\varphi}\) Oracles \(\hat{\varphi}(\alpha,\phi,a) \equiv \psi(\alpha,\phi,a) \text{ Used In The Functions } \hat{\text{R}}_{\alpha} \\ \hat{\varphi}(1,\alpha_1,\phi,a) \equiv \hat{\varepsilon}(\alpha_1,\phi,a) \text{ Used In The Functions } \hat{\text{R}}_{\alpha,\alpha_1}\forall\alpha \\ \hat{\varphi}(2,\alpha_1,\phi,a) \equiv \hat{\zeta}(\alpha_1,\phi,a) \text{ Used In The Functions } \hat{\text{R}}_{\alpha,\alpha_1,2}\forall\alpha \\ \hat{\varphi}(3,\alpha_1,\phi,a) \equiv \hat{\eta}(\alpha_1,\phi,a) \text{ Used In The Functions } \hat{\text{R}}_{\alpha,\alpha_1,3}\forall\alpha \\ \hat{\varphi}(\beta,2,\phi,a) \text{ acts as truth predicate for languages including } \hat{\varphi}(\beta-1,\alpha_1,\phi,a)\forall\alpha_1 \\ \hat{\varphi}(\beta,\gamma,\phi,a) \text{ acts as truth predicate for languages including } \hat{\varphi}(\beta-1,\delta,\phi,a)\forall\delta<\gamma \\ \hat{\varphi}(\beta,2,\phi,a) \text{ acts as truth predicate for languages including } \hat{\varphi}(\gamma,2,\phi,a)\forall\gamma<\beta \) \(\hat{\Gamma}\) Oracles This predicate is the limit of \(\hat{\varphi}(\beta,\alpha_1,\phi,a)\) for any \(\beta\). It also represents the limit of 4-argument \(\hat{\varphi}\) oracle notation. These are used in \(\hat{\text{R}}_{\alpha,\alpha_1,1,2}\) functions. These start at \(\hat{\Gamma}_2\). Extended \(\hat{\varphi}\) Notation \(\hat{\varphi}(2,1,\alpha_1,\phi,a) \equiv \hat{\Gamma}(\alpha_1,\phi,a) \text{ Used In The Functions } \hat{\text{R}}_{\alpha,\alpha_1,1,2}\forall\alpha \\ \hat{\varphi}(\cdots,\alpha_3,\alpha_2,\alpha_1,\phi,a) \text{ Used In The Functions } \hat{\text{R}}_{\alpha,\alpha_1,\alpha_2,\alpha_3,\cdots}\forall\alpha \\ \hat{\varphi}(1,\cdots,\alpha_2,\alpha_1,\phi,a) \equiv \hat{\varphi}(\cdots,\alpha_2,\alpha_1,\phi,a) \\ \hat{\varphi}(\cdots,\alpha_m,1,\cdots,\alpha_2,\alpha_1,\phi,a) \text{ acts as truth predicate for languages including } \\ \quad \hat{\varphi}(\cdots,\alpha,\cdots,\alpha_2,\alpha_1,\phi,a)\forall\alpha? \\ \hat{\varphi}(\cdots,\alpha_2,2,\phi,a) \text{ acts as truth predicate for languages including } \\ \quad \hat{\varphi}(\cdots,\alpha_2-1,\alpha_1,\phi,a)\forall\alpha_1 \\ \hat{\varphi}(\cdots,\alpha_2,\gamma,\phi,a) \text{ acts as truth predicate for languages including } \\ \quad \hat{\varphi}(\cdots,\alpha_2-1,\delta,\phi,a)\forall\delta<\gamma \\ \hat{\varphi}(\cdots,\alpha_2,2,\phi,a) \text{ acts as truth predicate for languages including } \\ \quad \hat{\varphi}(\cdots,\gamma,2,\phi,a)\forall\gamma<\alpha_2 \\ \hat{\varphi}(\cdots,\alpha_m,1,\cdots,\alpha_1,\phi,a) \text{ acts as truth predicate for languages including } \\ \quad \hat{\varphi}(\cdots,\alpha_2-1,\alpha_1,\phi,a)\forall\alpha_1\) \(\not\varphi\) Oracle This is the most powerful truth predicate defined here. This is currently defined as \(\not\varphi(n,\phi,a) \equiv \hat{\varphi}(\underbrace{\cdots,\alpha,\alpha,\alpha}_n,\phi,a)\forall\alpha\), however this definition may be changed in the future. The \(\not\varphi\) oracles can be used in set-theoretic language when Godel-Encoded and represented in Polish Notation. This oracle is used as part of the language in the \(\hat{\text{R}}^{\not\varphi}\) function. Category:Potentially ILL-Defined Category:C7X's Stuff Category:Doubly Uncomputable Numbers Category:SALAD NUMBERS Category:Largest Googolisms Throughout History